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Assume you have a resistor and an inductance in series. When a DC supply is connected at t=0, the coil is inducing an opposing voltage, which is the same as the DC supply's voltage. The current through the circuit at this time is zero.

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But how can the coil at t=0 induce an opposing voltage without current? My understanding is that you need a changing magnetic field to induce a voltage, for which you need current. But at t=0 there is no current, so how does coil induce an opposing voltage?

Please help me out as I want to have deep understanding of circuits.

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  • $\begingroup$ It states right in the problem, a DC source is connected at $t=0$, so there is indeed a current. $\endgroup$
    – Triatticus
    Commented Jan 1, 2021 at 23:54
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    $\begingroup$ Current at t=0 is indeed zero but the derivative of current is > 0. Induced emf is proportional to derivative of current. $\endgroup$
    – R. Emery
    Commented Jan 2, 2021 at 0:24

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Coil is not "inducing voltage", rather there is induced EMF being generated. Mobile charge carriers in the coil produce induced electric field, whose net total effect is quantified by so-called EMF (electromotive force).

Magnitude of this EMF is, due to Faraday's law, given by $$ EMF = -L\frac{dI}{dt} $$

The important thing is how quickly current changes, not its value. In other words, if current $I$ is zero but its derivative $dI/dt$ is not (mobile charge carriers are accelerating from zero speed), EMF is present. There is no need for current or magnetic field to be present.

The induced electric field is produced whenever the mobile charge carriers accelerate. Whether they have big speed is irrelevant.

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  • $\begingroup$ The formula for the EMF is the same as for the voltage across the inductor (albeit with a minus sign). Can you elaborate on the difference between EMF and voltage? Also I still dont quiet understand where the inductor is getting the energy from to create this EMF, if not from the DC supply via a current( which doesnt exist at the first instace) $\endgroup$
    – fihdi
    Commented Jan 2, 2021 at 12:31
  • $\begingroup$ Induced EMF in coil is integral of induced (solenoidal) field due to mobile charge carriers accelerating, integration being done for path that goes inside the coil conductor. Voltage is an overloaded term, but in circuit theory, it is drop of Coulomb potential when going from point 1 to point 2. For ideal coil it has the same magnitude as EMF because net electric field has to be zero in the coil conductor. For real coil the magnitude is lower than EMF magnitude, since some field is still present in the coil conductor to push charges against the ohmic resistance. $\endgroup$
    – Ján Lalinský
    Commented Jan 2, 2021 at 14:10
  • $\begingroup$ No energy is needed at $t=0$; EMF is quantification of effect of induced electric force, not of energy of that electric field. Some energy source is needed to sustain the EMF since at $t>0$, current and magnetic field energy are greater than zero. In circuits this energy is usually supplied by power source (in your example, the DC power supply) when current flows. So at $t=0$, energy delivered is zero but EMF is already at finite value and as time goes, both current, magnetic field and magnetic field energy increases. $\endgroup$
    – Ján Lalinský
    Commented Jan 2, 2021 at 14:18

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